November 27, 2007 2:08 PM

Conditional expectations Tuesday, November 27, 2007 2:08 PM Homework 4 to be posted by Wednesday morning, due Friday, December 7, absolute late deadline Tuesday, December 11 at 5 PM. In our calculation with the server problem from the last lecture we could decompose the random variable T as follows: Let's do another calculation on this problem: What is the expected value of the time until the second task is finished? Of course one way to do this is to simply integrate over the CDF for T which we just computed. But it will be useful to illustrate how to do this calculation directly without using conditional probability, but only conditional expectation because in some problems, it is sufficiently easy to calculate conditional expectations but not conditional probabilities. Conditional expectation of a random variable on an event: This definition agrees with the intuition that the conditional expectation of a random variable should be an average of advprobnotes112707b Page 1 pg the random variable with respect to the probability measure restricted to the outcomes consistent with the given event B. But how do we see that this is equivalent to integrating x against the conditional probability distribution? Note that: Law of Total Expectation: If we have a partition Returning to our problem: advprobnotes112707b Page 2 Returning to our problem: This argument can be generalized to show that if a random variable ( ) can be expressed in terms of some other random variables ( ) over the event being conditioned upon, one can replace the random variable ( ) by its representation ( ) valid over the conditioned event advprobnotes112707b Page 3 It is important to keep the condition! Continuing, This is equivalent to our formula for the definition of a conditional expectation if we think about the state space as being the same as the probability space, so our probability space is advprobnotes112707b Page 4 This is a result of general importance: It demonstrates the memory free property of the exponential distribution. In fact this is true not only of the expectation of exponential random variables but also for their probability distribution: advprobnotes112707b Page 5 This is the special memory-free property that characterizes exponential distributions (and not much else). Returning to our original calculation for the expected time for the second request to complete This should agree with integrating the probability distribution derived last lecture against t. Motivated by the desire to define conditional probabilities and expectations with respect to events of zero measure (i.e. that a continuous random variable takes a particular value), we go to a more abstract formulation which generalizes the above. General setup: Given probability space advprobnotes112707b Page 6 The sub sigma-algebra is simply a collection of measurable sets belonging to which form a sigma- algebra but need not be all of . To make these ideas more concrete, think of the followng example: Returning to our general discussion, we start by defining the conditional expectation (rather conditional probability) Of the random variable With respect to the sub sigma- algebra: advprobnotes112707b Page 7 Of the random variable With respect to the sub sigma algebra: This is a rather opaque definition but it's useful because it's completely general and one can directly show that it gives an essentially unique definition. (Radon-Nikodym Theorem). The non-uniqueness is only in that the conditional expectation can be changed on sets of measure zero. What is the intuitive interpretation of this conditional expectation with respect to a sub sigma algebra? Think of the sub sigma algebra as encoding partial information about the uncertainty. In other words, with the partial information, you can decide for each whether or not In particular, when then we can think about as encoding all information obtained by observing the random variable Y. advprobnotes112707b Page 8.

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